Homotopy \(G\)-algebra structure on Bredon-Illman cochain complex
Let \(G\) be a topological group and let \(X\) be a \(G\)-space. The orbit category \(\Pi _{G}( X) \) is defined to be the category whose objects are \(G\)-maps \(x_{H}:G/H\to X\) and whose morphisms have the form \(( \widehat{a},\omega ) :x_{H}\to x_{H^{\prime }}\), where \(a\in G\) satisfies \(a^{-1}Ha\subseteq H^{\prime }\), \(\widehat{a}:G/H\to G/H^{\prime }\) is defined by \(\widehat{a}( gH) =gaH^{\prime }\) and \(\omega \) is a \(G\)-homotopy from \(x_{H}\) to \(x_{H^{\prime }}\widehat{a}\) . Now, for \(\mathcal{R}\) being the category of commutative rings with 1, an equivariant local coefficient system on X is defined to be a contravariant functor \(M:\Pi _{G}( X) \to \mathcal{R}\). For any equivariant \(n\)-simplex \(\sigma :G/H\times \Delta ^{n}\to X\) we define the object \(\widetilde{\sigma }\)\ in the orbit category \(\Pi _{G}( X) \) by \(\widetilde{\sigma }( gH) =\sigma ( gH,e_{0}^{n}) \). The set of functions \(c\) defined on \(n\)-simplices with \(c( \sigma ) =M( \widetilde{\sigma }) \) is denoted by \(S_{G}^{n}( X,M) \) and has a ring structure induced by \(M\). The Bredon-Illman cochain complex is \(( S_{G}^{\ast }( X,M) ,\delta ) \) where the coboundary \(\delta \) is defined in the obvious way, after this the Bredon-Illman cohomology ring \( H_{G}^{\ast }( G,M) =H( S_{G}^{\ast }( X,M) ,\delta ) \) arrives naturally. This paper is devoted to showing in a fairly technical way that the Bredon-Illman cochain complex defined as above is a homotopy \(G\)-algebra in the category \(\mathbb{Z}\)-\textbf{mod} also having the structure of a differentially graded Lie algebra. The objects in the \(\mathbb{Z}\)-\textbf{mod} category are \(\mathbb{Z}\)-modules and the morphisms the corresponding module homomorphisms. A graded object \(H=\oplus H^{j}\) in this category is said to be a non-\(\Sigma \) operad provided the existence of graded product maps of degree 0. The first step in the proof of the main result is to prove that the Bredon-Illman cochain complex is a non-\(\Sigma \) operad in \(\mathbb{ Z}\)-\textbf{mod}. As a consequence follows the \(B_{\infty }\) -algebra structure for the Bredon-Illman cochain complex.
scientific article; zbMATH DE number 2116783